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Theorem elsncg 3755
 Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
elsncg (A V → (A {B} ↔ A = B))

Proof of Theorem elsncg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . 2 (x = A → (x = BA = B))
2 df-sn 3741 . 2 {B} = {x x = B}
31, 2elab2g 2987 1 (A V → (A {B} ↔ A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741 This theorem is referenced by:  elsnc  3756  elsni  3757  snidg  3758  eltpg  3769  eldifsn  3839  opkth1g  4130
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